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Theorem undir 3701
Description: Distributive law for union over intersection. Theorem 29 of [Suppes] p. 27. (Contributed by NM, 30-Sep-2002.)
Assertion
Ref Expression
undir  |-  ( ( A  i^i  B )  u.  C )  =  ( ( A  u.  C )  i^i  ( B  u.  C )
)

Proof of Theorem undir
StepHypRef Expression
1 undi 3699 . 2  |-  ( C  u.  ( A  i^i  B ) )  =  ( ( C  u.  A
)  i^i  ( C  u.  B ) )
2 uncom 3589 . 2  |-  ( ( A  i^i  B )  u.  C )  =  ( C  u.  ( A  i^i  B ) )
3 uncom 3589 . . 3  |-  ( A  u.  C )  =  ( C  u.  A
)
4 uncom 3589 . . 3  |-  ( B  u.  C )  =  ( C  u.  B
)
53, 4ineq12i 3641 . 2  |-  ( ( A  u.  C )  i^i  ( B  u.  C ) )  =  ( ( C  u.  A )  i^i  ( C  u.  B )
)
61, 2, 53eqtr4i 2443 1  |-  ( ( A  i^i  B )  u.  C )  =  ( ( A  u.  C )  i^i  ( B  u.  C )
)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1407    u. cun 3414    i^i cin 3415
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-v 3063  df-un 3421  df-in 3423
This theorem is referenced by:  undif1  3849  dfif4  3902  dfif5  3903  bwth  20205
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